Jianxi Li

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On the second largest Laplacian eigenvalues of graphs

2012-12-11

Jianxi Li , Ji-Ming Guo , Wai Chee Shiu

On the laplacian estrada index of a graph

2009-01-01

Jianxi Li , Chee Shiu , An Chang

Let G be a graph of order n. Let ?1 , ?2 , . . . , ?n be the eigenvalues of the adjacency matrix of G, and let ?1 … Let G be a graph of order n. Let ?1 , ?2 , . . . , ?n be the eigenvalues of the adjacency matrix of G, and let ?1 , ?2 , . . . , ?n be the eigenvalues of the Laplacian matrix of G. Much studied Estrada index of the graph G is defined n as EE = EE(G)= ?n/i=1 e?i . We define and investigate the Laplacian Estrada index of the graph G, LEE=LEE(G)= ?n/i=1 e(?i - 2m/n). Bounds for LEE are obtained, as well as some relations between LEE and graph Laplacian energy.

Bounds on normalized Laplacian eigenvalues of graphs

2014-08-21

Jianxi Li , Ji-Ming Guo , Wai Chee Shiu

Let G be a simple connected graph of order n, where . Its normalized Laplacian eigenvalues are . In this paper, some new upper and lower bounds on are obtained, … Let G be a simple connected graph of order n, where . Its normalized Laplacian eigenvalues are . In this paper, some new upper and lower bounds on are obtained, respectively. Moreover, connected graphs with (or ) are also characterized. MSC:05C50, 15A48.

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The harmonic index of a graph

2014-10-01

Jianxi Li , Wai Chee Shiu

The harmonic index of a graph $G$ is defined as the sum of weights $\frac{2}{d(v_i)+d(v_j)}$ of all edges $v_iv_j$ of $G$, where $d(v_i)$ denotes the degree of the vertex $v_i$ … The harmonic index of a graph $G$ is defined as the sum of weights $\frac{2}{d(v_i)+d(v_j)}$ of all edges $v_iv_j$ of $G$, where $d(v_i)$ denotes the degree of the vertex $v_i$ in $G$. In this paper, we study how the harmonic index behaves when the graph is under perturbations. These results are used to provide a simpler method for determining the unicyclic graphs with maximum and minimum harmonic index among all unicyclic graphs, respectively. Moreover, a lower bound for harmonic index is also obtained.

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On the Laplacian, signless Laplacian and normalized Laplacian characteristic polynomials of a graph

2013-09-01

Ji-Ming Guo , Jianxi Li , Wai Chee Shiu

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The Laplacian spectral radius of some graphs

2009-03-20

Jianxi Li , Wai Chee Shiu , Wai Hong Ronald Chan

The number of spanning trees of a graph

2009-10-25

Jianxi Li , Wai Chee Shiu , An Chang

Bounding the sum of powers of normalized Laplacian eigenvalues of a graph

2017-12-24

Jianxi Li , Ji‐Ming Guo , Wai Chee Shiu +2 more

On the nullity of bicyclic graphs

2008-01-01

Jianxi Li , An Chang , Wai Chee Shiu

The nullity of a graph is the multiplicity of the eigenvalues zero in its spectrum. In this papers, we obtain the nullity set of n−vertex bicyclic graphs, and characterize the … The nullity of a graph is the multiplicity of the eigenvalues zero in its spectrum. In this papers, we obtain the nullity set of n−vertex bicyclic graphs, and characterize the bicyclic graphs with maximal nullity.

An edge-separating theorem on the second smallest normalized Laplacian eigenvalue of a graph and its applications

2014-03-21

Jianxi Li , Ji-Ming Guo , Wai Chee Shiu +1 more

Let λ2(G) be the second smallest normalized Laplacian eigenvalue of a graph G. In this paper, we investigate the behavior on λ2(G) when the graph G is perturbed by separating … Let λ2(G) be the second smallest normalized Laplacian eigenvalue of a graph G. In this paper, we investigate the behavior on λ2(G) when the graph G is perturbed by separating an edge. This result can be used to determine all trees and unicyclic graphs with λ2(G)≥1−22. Moreover, the trees and unicyclic graphs with λ2(G)=1−22 are also determined, respectively.

The smallest values of algebraic connectivity for unicyclic graphs

2010-06-29

Jianxi Li , Ji‐Ming Guo , Wai Chee Shiu

The harmonic index of unicyclic graphs with given matching number

2014-01-01

L.V. Jian-Bo , Jianxi Li , Shiu Chee

The harmonic index of a graph G is defined as the sum of weights 2/ d(u)+d(v) of all edges uv of G, where d(u) and d(v) are the degrees of … The harmonic index of a graph G is defined as the sum of weights 2/ d(u)+d(v) of all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G, respectively. In this paper, we determine the graph with minimum harmonic index among all unicyclic graphs with a perfect matching. Moreover, the graph with minimum harmonic index among all unicyclic graphs with a given matching number is also determined.

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The algebraic connectivity of lollipop graphs

2011-01-18

Ji-Ming Guo , Wai Chee Shiu , Jianxi Li

The Harmonic Index of Some Graphs

2015-12-11

Jianxi Li , Jian-Bo Lv , Yang Liu

List injective edge-coloring of subcubic graphs

2021-07-14

Jian-Bo Lv , Jianxi Li , Nian Hong Zhou

The Hosoya indices and Merrifield–Simmons indices of graphs with connectivity at most <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi></mml:math>

2011-10-02

Kexiang Xu , Jianxi Li , Lingping Zhong

Some results on the Laplacian eigenvalues of unicyclic graphs

2009-01-04

Jianxi Li , Wai Chee Shiu , Wai Hong Ronald Chan

Some Properties on the Harmonic Index of Molecular Trees

2014-01-29

Shaoqiang Liu , Jianxi Li

The harmonic index of a graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math> is defined as the sum of weights <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mi>d</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfenced></mml:mrow></mml:mrow></mml:math> of all edges <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" … The harmonic index of a graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math> is defined as the sum of weights <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mi>d</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfenced></mml:mrow></mml:mrow></mml:math> of all edges <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mi>uv</mml:mi></mml:mrow></mml:math> of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mi>d</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math> denotes the degree of the vertex <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mrow><mml:mi>u</mml:mi></mml:mrow></mml:math> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math>. In this paper, some general properties of the harmonic index for molecular trees are explored. Moreover, the smallest and largest values of harmonic index for molecular trees with given pendent vertices are provided, respectively.

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Six classes of trees with largest normalized algebraic connectivity

2014-04-23

Jianxi Li , Ji-Ming Guo , Wai Chee Shiu +1 more

The normalized algebraic connectivity of a graph G, denoted by λ2(G), is the second smallest eigenvalue of its normalized Laplacian matrix. In this paper, we firstly determine all trees with … The normalized algebraic connectivity of a graph G, denoted by λ2(G), is the second smallest eigenvalue of its normalized Laplacian matrix. In this paper, we firstly determine all trees with λ2(G)⩾1−63. Then we classify such trees into six classes C1,…,C6 and prove that λ2(Ti)>λ2(Tj) for 1⩽i<j⩽6, where Ti∈Ci and Tj∈Cj. At the same time, the values of the normalized algebraic connectivity for the six classes of trees are provided, respectively. These results are similar to those on the algebraic connectivity which were obtained by Yuan et al. (2008) [8].

The Largest Normalized Laplacian Spectral Radius of Non-Bipartite Graphs

2015-11-19

Ji-Ming Guo , Jianxi Li , Wai Chee Shiu

Graphs <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e142" altimg="si34.svg"><mml:mi>G</mml:mi></mml:math> with nullity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e147" altimg="si35.svg"><mml:mrow><mml:mn>2</mml:mn><mml:mi>c</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>p</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</…

2022-02-04

Sarula Chang , Bit-Shun Tam , Jianxi Li +1 more

A note on the upper bounds for the Laplacian spectral radius of graphs

2013-06-04

Ji-Ming Guo , Jianxi Li , Wai Chee Shiu

Note on the Laplacian Estrada index of a graph

2011-01-01

Jianxi Li , Wai Chee Shiu , Wai Hong Ronald Chan

Let G be a simple graph of order n with m edges. The Laplacian Estrada index of G is defined as LEE = LEE(G )= n i=1 e Let G be a simple graph of order n with m edges. The Laplacian Estrada index of G is defined as LEE = LEE(G )= n i=1 e

On the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math>th Laplacian eigenvalues of trees with perfect matchings

2009-11-05

Jianxi Li , Wai Chee Shiu , An Chang

The Laplacian spectral radius of graphs

2010-08-02

Jianxi Li , Wai Chee Shiu , An Chang

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Effects on the normalized Laplacian spectral radius of non-bipartite graphs under perturbation and their applications

2016-02-10

Ji-Ming Guo , Jianxi Li , Wai Chee Shiu

The normalized Laplacian eigenvalues of a network play an important role in its structural and dynamical aspects associated with the network. In this paper, we consider how the normalized Laplacian … The normalized Laplacian eigenvalues of a network play an important role in its structural and dynamical aspects associated with the network. In this paper, we consider how the normalized Laplacian spectral radius of a non-bipartite graph behaves by several graph operations. As an example of the application, the smallest normalized Laplacian spectral radius of non-bipartite unicyclic graphs with fixed order is determined.

On the harmonic index and the matching number of a tree.

2014-01-01

Jian-Bo Lv , Jianxi Li

The harmonic index of a graph G is defined as the sum of weights 2 d(u)+d(v) of all edges uv of G, where d(u) and d(v) are the degrees of … The harmonic index of a graph G is defined as the sum of weights 2 d(u)+d(v) of all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G, respectively. In this paper, we give a sharp lower bound on the harmonic index of trees with a perfect matching in terms of the number of vertices. A sharp lower bound on the harmonic index of trees with a given size of matching is also obtained.

The harmonic index of a graph and its DP-chromatic number

2020-04-10

Jian-Bo Lv , Jianxi Li

Maximizing the A-spectral radius of graphs with given size and diameter

2022-06-20

Peng Huang , Jianxi Li , Wai Chee Shiu

The minimum algebraic connectivity of caterpillar unicyclic graphs

2011-01-01

Wai Chee Shiu , Ji-Ming Guo , Jianxi Li

A caterpillar unicyclic graph is a unicyclic graph in which the removal of all pendant vertices makes it a cycle. In this paper, the unique caterpillar unicyclic graph with minimum … A caterpillar unicyclic graph is a unicyclic graph in which the removal of all pendant vertices makes it a cycle. In this paper, the unique caterpillar unicyclic graph with minimum algebraic connectivity among all caterpillar unicyclic graphs is determined.

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Coefficients of the Characteristic Polynomial of the (Signless, Normalized) Laplacian of a Graph

2017-07-21

Ji-Ming Guo , Jianxi Li , Peng Huang +1 more

The Orderings of Bicyclic Graphs and Connected Graphs by Algebraic Connectivity

2010-12-03

Jianxi Li , Ji-Ming Guo , Wai Chee Shiu

The algebraic connectivity of a graph $G$ is the second smallest eigenvalue of its Laplacian matrix. Let $\mathscr{B}_n$ be the set of all bicyclic graphs of order $n$. In this … The algebraic connectivity of a graph $G$ is the second smallest eigenvalue of its Laplacian matrix. Let $\mathscr{B}_n$ be the set of all bicyclic graphs of order $n$. In this paper, we determine the last four bicyclic graphs (according to their smallest algebraic connectivities) among all graphs in $\mathscr{B}_n$ when $n\geq 13$. This result, together with our previous results on trees and unicyclic graphs, can be used to further determine the last sixteen graphs among all connected graphs of order $n$. This extends the results of Shao et al. [The ordering of trees and connected graphs by their algebraic connectivity, Linear Algebra Appl. 428 (2008) 1421-1438].

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On the sum of the two largest Laplacian eigenvalues of unicyclic graphs

2015-09-17

Yirong Zheng , An Chang , Jianxi Li

Let G be a simple graph and $S_{2}(G)$ be the sum of the two largest Laplacian eigenvalues of G. Guan et al. (J. Inequal. Appl. 2014:242, 2014) determined the largest … Let G be a simple graph and $S_{2}(G)$ be the sum of the two largest Laplacian eigenvalues of G. Guan et al. (J. Inequal. Appl. 2014:242, 2014) determined the largest value for $S_{2}(T)$ among all trees of order n. They also conjectured that among all connected graphs of order n with m ( $n \leq m\leq2n -3$ ) edges, $G_{m,n}$ is the unique graph which has maximal value of $S_{2}(G)$ , where $G_{m,n}$ is a graph of order n with m edges which has $m-n+1$ triangles with a common edge and $2n-m-3$ pendent edges incident with one end vertex of the common edge. In this paper, we confirm the conjecture with $m=n$ .

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Maximizing the signless Laplacian spectral radius of k-connected graphs with given diameter

2021-01-30

Peng Huang , Jianxi Li , Wai Chee Shiu

The irregularity of two types of trees

2016-06-08

Jianxi Li , Yang Liu , Wai Chee Shiu

The irregularity of a graph $G$ is defined as the sum of weights $|d(u)-d(v)|$ of all edges $uv$ of $G$, where $d(u)$ and $d(v)$ are the degrees of the vertices … The irregularity of a graph $G$ is defined as the sum of weights $|d(u)-d(v)|$ of all edges $uv$ of $G$, where $d(u)$ and $d(v)$ are the degrees of the vertices $u$ and $v$ in $G$, respectively. In this paper, some structural properties on trees with maximum (or minimum) irregularity among trees with given degree sequence and trees with given branching number are explored, respectively. Moreover, the corresponding trees with maximum (or minimum) irregularity are also found, respectively.

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The smallest Laplacian spectral radius of graphs with a given clique number

2012-05-04

Ji-Ming Guo , Jianxi Li , Wai Chee Shiu

Bicyclic graphs with maximum sum of the two largest Laplacian eigenvalues

2016-11-18

Yirong Zheng , An Chang , Jianxi Li +1 more

Let G be a simple connected graph and $S_{2}(G)$ be the sum of the two largest Laplacian eigenvalues of G. In this paper, we determine the bicyclic graph with maximum … Let G be a simple connected graph and $S_{2}(G)$ be the sum of the two largest Laplacian eigenvalues of G. In this paper, we determine the bicyclic graph with maximum $S_{2}(G)$ among all bicyclic graphs of order n, which confirms the conjecture of Guan et al. (J. Inequal. Appl. 2014:242, 2014) for the case of bicyclic graphs.

Graphs G with nullity n(G)−g(G)−1

2022-02-28

Sarula Chang , Jianxi Li

$$\ell $$-Connectivity, Integrity, Tenacity, Toughness and Eigenvalues of Graphs

2022-09-02

Hongzhang Chen , Jianxi Li

On the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e115" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub></mml:math>-spectral radius of graphs with given size

2023-08-08

Hongzhang Chen , Jianxi Li , Peng Huang

The Randic Indices of Trees, Unicyclic Graphs and Bicyclic Graphs.

2016-01-01

Jianxi Li , Selvaraj Balachandran , S. K. Ayyaswamy +1 more

Approaches Which Output Infinitely Many Graphs With Small Local Antimagic Chromatic Number

2020-01-01

Gee-Choon Lau , Jianxi Li , Ho-Kuen Ng +1 more

An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair … An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $χ_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, we (i) give a sufficient condition for a graph with one pendant to have $χ_{la}\ge 3$. A necessary and sufficient condition for a graph to have $χ_{la}=2$ is then obtained; (ii) give a sufficient condition for every circulant graph of even order to have $χ_{la} = 3$; (iii) construct infinitely many bipartite and tripartite graphs with $χ_{la} = 3$ by transformation of cycles; (iv) apply transformation of cycles to obtain infinitely many one-point union of regular (possibly circulant) or bi-regular graphs with $χ_{la} = 2,3$. The work of this paper suggests many open problems on the local antimagic chromatic number of bipartite and tripartite graphs.

On Extremal Graphs for Zero Forcing Number

2022-11-07

Yi-Ping Liang , Jianxi Li , Shou‐Jun Xu

NOTES ON THE HARMONIC INDEX OF GRAPHS

2022-12-01

Yirong Zheng , Jian-Bo Lv , Jianxi Li

For a connected graph G of order n, the harmonic index of G is the sum of weights 2d(u)+d(v) over all edges uv of G, where d(u) and d(v) are … For a connected graph G of order n, the harmonic index of G is the sum of weights 2d(u)+d(v) over all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G, respectively. We prove that H(G)≥χDP(G)−22+2(χDP(G)−1)n+χDP(G)−2+2(n−χDP(G))n, and this bound is sharp for all n and 2≤χDP(G)≤n, where χDP(G) is the DP-chromatic number of G. This generalizes the previous lower bounds on H(G). Moreover, we also determine the tree with minimum harmonic index among trees in 𝒯n,l, where 𝒯n,l is the set of trees of order n with a given segment sequence l.

Spectral bounds for the zero forcing number of a graph

2023-01-29

Hongzhang Chen , Jianxi Li , Shou‐Jun Xu

Let Z(G) be the zero forcing number of a simple connected graph G.In this paper, we study the relationship between the zero forcing number of a graph and its (normalized) … Let Z(G) be the zero forcing number of a simple connected graph G.In this paper, we study the relationship between the zero forcing number of a graph and its (normalized) Laplacian eigenvalues.We provide the upper and lower bounds on Z(G) in terms of its (normalized) Laplacian eigenvalues, respectively.Our bounds extend the existing bounds for regular graphs.

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Sufficient conditions for component factors in a graph

2024-03-22

Hongzhang Chen , Xiaoyun Lv , Jianxi Li

Some results on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e44" altimg="si18.svg"><mml:mrow><mml:mo>{</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>i</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>:</mml:mo><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="…

2024-09-06

Xiaoyun Lv , Jianxi Li , Shou‐Jun Xu

On the Laplacian spectral radii of bipartite graphs

2011-05-06

Jianxi Li , Wai Chee Shiu , Wai Hong Ronald Chan

On the spectral radius of unicyclic graphs with fixed girth

2013-01-01

Jianxi Li , Ji-Ming Guo , Wai Chee Shiu

The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Let U g n be the set of unicyclic graphs of order n with girth g. … The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Let U g n be the set of unicyclic graphs of order n with girth g. For all integers n and g with 5 ≤ g ≤ n − 6, we determine the first ⌊ g2⌋+ 3 spectral radii of unicyclic graphs in the set U g n .

The effect on eigenvalues of connected graphs by adding edges

2018-02-12

Ji-Ming Guo , Pan-Pan Tong , Jianxi Li +2 more

The $$A_{\alpha }$$-spectral radius for $$\{P_{2}, C_{3}, P_{5}, \mathcal {T}(3)\}$$-factors in graphs

2025-05-06

Xiaoyun Lv , Jianxi Li , Shou‐Jun Xu

Characterizing path-factor deleted graphs via $Q$-index and $\mathcal{D}$-index

2025-03-17

Xiaoyun Lv , Jianxi Li , Shou‐Jun Xu

A graph $G$ has a $\mathcal{P}_{\geq k}$-factor if $G$ has a spanning subgraph $H$ such that every component of $H$ is a path of order at least $k$. A graph … A graph $G$ has a $\mathcal{P}_{\geq k}$-factor if $G$ has a spanning subgraph $H$ such that every component of $H$ is a path of order at least $k$. A graph $G$ is $\mathcal{P}_{\geq k}$-factor deleted if $G-e$ has a $\mathcal{P}_{\geq k}$-factor for each edge $e \in E(G)$. In this paper, we study the $\mathcal{P}_{\geq 2}$-factor deleted graphs by their $Q$-index (the largest eigenvalue of the signless Laplacian matrix) and $\mathcal{D}$-index (the largest eigenvalue of the distance matrix). Sufficient conditions (in terms of $Q$-index and $\mathcal{D}$-index) to guarantee that a graph $G$ to be a $\mathcal{P}_{\geq 2}$-factor deleted graph are established.

Some results on the total (zero) forcing number of a graph

2025-03-10

Jianxi Li , Dongxin Tu , Wai Chee Shiu

Two Variants of Toughness of a Graph and its Eigenvalues

2025-03-01

Hongzhang Chen , Jianxi Li , Shou‐Jun Xu

The $$A_\alpha $$-spectral radius of graphs with chorded cycles

2024-12-10

Longzhe Jin , Jianxi Li , Peng Huang

Clique trees with a given zero forcing number maximizing the $$A_\alpha $$-spectral radius

2024-10-20

Longzhe Jin , Jianxi Li , Yuan Hou

Some results on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e44" altimg="si18.svg"><mml:mrow><mml:mo>{</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>i</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>:</mml:mo><mml:mi>i</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="…

2024-09-06

Xiaoyun Lv , Jianxi Li , Shou‐Jun Xu

The eigenvalue multiplicity of line graphs

2024-09-03

Sarula Chang , Jianxi Li , Yirong Zheng

The second smallest normalized Laplacian eigenvalue of unicyclic graphs

2024-06-17

Ji‐Ming Guo , E. K. S. LIU , Jianxi Li +1 more

The Zero (Total) Forcing Number and Covering Number of Trees

2024-03-26

Dongxin Tu , Jianxi Li , Wai Chee Shiu

Let $F(G)$, $F_{t}(G)$, $\beta(G)$, and $\beta'(G)$ be the zero forcing number, the total forcing number, the vertex covering number and the edge covering number of a graph $G$, respectively. In … Let $F(G)$, $F_{t}(G)$, $\beta(G)$, and $\beta'(G)$ be the zero forcing number, the total forcing number, the vertex covering number and the edge covering number of a graph $G$, respectively. In this paper, we first completely characterize all trees $T$ with $F(T) = (\Delta-2) \beta(T) + 1$, solving a problem proposed by Brimkov et al. in 2023. Next, we study the relationship between the zero (or total) forcing number of a tree and its edge covering number, and show that $F(T) \leq \beta'(T)-1$ and $F_{t}(T) \leq \beta'(T)$ for any tree $T$ of order $n \geq 3$. Moreover, we also characterize all trees $T$ with $F(T) = \beta'(T)-1$ and $F(T) = \beta'(T)-2$, respectively.

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A characterization on trees $T$ with $m(T, \lambda)=p(T)-2$

2024-03-26

Sarula Chang , Jianxi Li , Yirong Zheng

Let $m(G,\lambda)$ be the multiplicity of an eigenvalue $\lambda$ of a connected graph $G$. Wang et al. [Linear Algebra Appl. 584(2020), 257-266] proved that for any connected graph $G\neq C_n$, … Let $m(G,\lambda)$ be the multiplicity of an eigenvalue $\lambda$ of a connected graph $G$. Wang et al. [Linear Algebra Appl. 584(2020), 257-266] proved that for any connected graph $G\neq C_n$, $m(G, \lambda) \leq 2c(G) + p(G) -1$, where $c (G) = |E(G)| - |V (G)| + 1$ and $p(G)$ are the cyclomatic number and the number of pendant vertices of $G$, respectively. In the same paper, they proposed the problem to characterize all connected graphs $G$ with eigenvalue $\lambda$ such that $m(G, \lambda) =2c (G)+ p(G)-1$. Wong et al. [Discrete Math. 347(2024), 113845] solved this problem for the case when $G$ is a tree by characterizing all trees $T$ with eigenvalue $\lambda$ such that $m(T , \lambda) = p(T )-1$. In this paper, we further provide the structural characterization on trees $T$ with eigenvalue $\lambda$ such that $m(T , \lambda) = p(T )-2$.

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A characterization on trees T with m(T, λ) = p(T ) -2

2024-03-26

Sarula Chang , Jianxi Li , Yirong Zheng

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Sufficient conditions for component factors in a graph

2024-03-22

Hongzhang Chen , Xiaoyun Lv , Jianxi Li

Spectral bounds for the vulnerability parameters of graphs

2024-02-13

Hongzhang Chen , Jianxi Li , Wai Chee Shiu

BOUNDING THE Aα-SPECTRAL RADIUS OF k-CONNECTED IRREGULAR GRAPHS

2024-02-01

Jianxi Li , Hongzhang Chen , Peng Huang

Let G be a simple graph of order n. For α∈[0,1], the Aα-matrix of G is defined as Aα=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and the diagonal … Let G be a simple graph of order n. For α∈[0,1], the Aα-matrix of G is defined as Aα=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and the diagonal matrix of vertex degrees of G, respectively. The largest eigenvalue of Aα(G), denoted by ρα(G), is called the Aα spectral radius of G. We give an upper bound on ρα(G) for k-connected irregular graphs. Moreover, we also derive an upper bound on ρα(G) when G is a subgraph of a k-connected regular graph. Our results improve or extend the existing results, respectively.

Partition line graphs of multigraphs into two subgraphs with large chromatic numbers

2023-12-08

Jian-Bo Lv , Jianxi Li

On the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e115" altimg="si24.svg"><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub></mml:math>-spectral radius of graphs with given size

2023-08-08

Hongzhang Chen , Jianxi Li , Peng Huang

Remarks on the Largest Eigenvalue of a Signed Graph

2023-07-10

Kaiyang Lan , Jianxi Li , Feng Liu

Nullities of cycle-spliced bipartite graphs

2023-06-10

Sarula Chang , Jianxi Li , Yirong Zheng

For a simple graph $G$, let $\eta(G)$ and $c(G)$ be the nullity and the cyclomatic number of $G$, respectively. A cycle-spliced bipartite graph is a connected graph in which every … For a simple graph $G$, let $\eta(G)$ and $c(G)$ be the nullity and the cyclomatic number of $G$, respectively. A cycle-spliced bipartite graph is a connected graph in which every block is an even cycle. It was shown by Wong et al. (2022) that for every cycle-spliced bipartite graph $G$, $0\leq\eta(G)\leq c(G)+1$. Moreover, the extremal graphs with $\eta(G) = c(G)+1$ and $\eta(G) =0$, respectively, have been characterized. In this paper, we prove that there is no cycle-spliced bipartite graphs $G$ of any order with nullity $\eta(G)=c(G)$. Furthermore, we also provide a structural characterization on cycle-spliced bipartite graphs $G$ with nullity $\eta(G)=c(G)-1$.

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Spectral bounds for the zero forcing number of a graph

2023-01-29

Hongzhang Chen , Jianxi Li , Shou‐Jun Xu

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NOTES ON THE HARMONIC INDEX OF GRAPHS

2022-12-01

Yirong Zheng , Jian-Bo Lv , Jianxi Li

On Extremal Graphs for Zero Forcing Number

2022-11-07

Yi-Ping Liang , Jianxi Li , Shou‐Jun Xu

Some results on the A α -eigenvalues of a graph

2022-10-18

Hongzhang Chen , Jianxi Li , Wai Chee Shiu

Let G be a simple graph of order n. For α∈[0,1], the Aα-matrix of G is defined as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and the diagonal … Let G be a simple graph of order n. For α∈[0,1], the Aα-matrix of G is defined as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and the diagonal degree matrix of G, respectively. The eigenvalues of the Aα-matrix of G are called the Aα-eigenvalues of G. In this paper, we first study the properties on Aα-eigenvalues, i.e. how the Aα-eigenvalues behave under some kinds of graph transformations including vertex deletion, vertex contraction, edge deletion and edge subdivision. Moreover, we also present the relationships between the Aα-eigenvalues of G and its k-domination number, independence number, chromatic number and circumference, respectively.

$$\ell $$-Connectivity, Integrity, Tenacity, Toughness and Eigenvalues of Graphs

2022-09-02

Hongzhang Chen , Jianxi Li

Trees with maximum sum of the two largest Laplacian eigenvalues

2022-07-15

Yirong Zheng , Jianxi Li , Sarula Chang

Let $T$ be a tree of order $n$ and $S_2(T)$ be the sum of the two largest Laplacian eigenvalues of $T$. Fritscher et al. proved that for any tree $T$ … Let $T$ be a tree of order $n$ and $S_2(T)$ be the sum of the two largest Laplacian eigenvalues of $T$. Fritscher et al. proved that for any tree $T$ of order $n$, $S_2(T) \leq n+2-\frac{2}{n}$. Guan et al. determined the tree with maximum $S_2(T)$ among all trees of order $n$. In this paper, we characterize the trees with $S_2(T) \geq n+1$ among all trees of order $n$ except some trees. Moreover, among all trees of order $n$, we also determine the first $\lfloor\frac{n-2}{2}\rfloor$ trees according to their $S_2(T)$. This extends the result of Guan et al.

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Maximizing the A-spectral radius of graphs with given size and diameter

2022-06-20

Peng Huang , Jianxi Li , Wai Chee Shiu

Upper bound for DP-chromatic number of a graph

2022-04-22

Jian-Bo Lv , Jianxi Li , Ziwen Huang

Graphs G with nullity n(G)−g(G)−1

2022-02-28

Sarula Chang , Jianxi Li

Graphs <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e142" altimg="si34.svg"><mml:mi>G</mml:mi></mml:math> with nullity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e147" altimg="si35.svg"><mml:mrow><mml:mn>2</mml:mn><mml:mi>c</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:mi>p</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</…

2022-02-04

Sarula Chang , Bit-Shun Tam , Jianxi Li +1 more

Extremal values for the variation of the Randić index of bicyclic graphs

2021-08-01

Jian-Bo Lv , Jianxi Li

Let G be a connected graph of order n. The variation of the Randić index of G is defined as R′(G)= ∑uv∈E(G)1max{d(u),d(v)}, where the summation goes over all edges uv … Let G be a connected graph of order n. The variation of the Randić index of G is defined as R′(G)= ∑uv∈E(G)1max{d(u),d(v)}, where the summation goes over all edges uv of G and d(u) is the degree of the vertex u in G. In this paper, among all bicyclic graphs of order n, the minimum and maximum values for R′ are determined, respectively.

List injective edge-coloring of subcubic graphs

2021-07-14

Jian-Bo Lv , Jianxi Li , Nian Hong Zhou

Maximizing the signless Laplacian spectral radius of k-connected graphs with given diameter

2021-01-30

Peng Huang , Jianxi Li , Wai Chee Shiu

The harmonic index of a graph and its DP-chromatic number

2020-04-10

Jian-Bo Lv , Jianxi Li

Approaches Which Output Infinitely Many Graphs With Small Local Antimagic Chromatic Number

2020-01-01

Gee-Choon Lau , Jianxi Li , Ho-Kuen Ng +1 more

The effect on eigenvalues of connected graphs by adding edges

2018-02-12

Ji-Ming Guo , Pan-Pan Tong , Jianxi Li +2 more

Bounding the sum of powers of normalized Laplacian eigenvalues of a graph

2017-12-24

Jianxi Li , Ji‐Ming Guo , Wai Chee Shiu +2 more

Coefficients of the Characteristic Polynomial of the (Signless, Normalized) Laplacian of a Graph

2017-07-21

Ji-Ming Guo , Jianxi Li , Peng Huang +1 more

Bicyclic graphs with maximum sum of the two largest Laplacian eigenvalues

2016-11-18

Yirong Zheng , An Chang , Jianxi Li +1 more

The irregularity of two types of trees

2016-06-08

Jianxi Li , Yang Liu , Wai Chee Shiu

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Effects on the normalized Laplacian spectral radius of non-bipartite graphs under perturbation and their applications

2016-02-10

Ji-Ming Guo , Jianxi Li , Wai Chee Shiu

The Randic Indices of Trees, Unicyclic Graphs and Bicyclic Graphs.

2016-01-01

Jianxi Li , Selvaraj Balachandran , S. K. Ayyaswamy +1 more

The Harmonic Index of Some Graphs

2015-12-11

Jianxi Li , Jian-Bo Lv , Yang Liu

The Largest Normalized Laplacian Spectral Radius of Non-Bipartite Graphs

2015-11-19

Ji-Ming Guo , Jianxi Li , Wai Chee Shiu

On the sum of the two largest Laplacian eigenvalues of unicyclic graphs

2015-09-17

Yirong Zheng , An Chang , Jianxi Li

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The harmonic index of a graph

2014-10-01

Jianxi Li , Wai Chee Shiu

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The Spectral Radii of Graphs with Prescribed Degree Sequence

2014-09-23

Jianxi Li , Wai Chee Shiu

In this paper, we first present the properties of the graph which maximize the spectral radius among all graphs with prescribed degree sequence. Using these results, we provide a somewhat … In this paper, we first present the properties of the graph which maximize the spectral radius among all graphs with prescribed degree sequence. Using these results, we provide a somewhat simpler method to determine the unicyclic graph with maximum spectral radius among all unicyclic graphs with a given degree sequence. Moreover, we determine the bicyclic graph which has maximum spectral radius among all bicyclic graphs with a given degree sequence. Let G be a simple connected graph with vertex set V (G) and edge set E(G). Its order is |V (G)|, denoted by n, and its size is |E(G)|, denoted by m. For v ∈ V (G), let NG(v) (or N (v) for short) be the set of all neighbors of v in G and let d(v) = |N (v)| be the degree of v. We use G − e and G + e to denote the graphs obtained by deleting the edge e from G and by adding the edge e to G, respectively. For any nonempty subset W of V (G), the subgraph of G induced by W is denoted by G(W ). The distance of u and v (in G) is the length of the shortest path between u and v, denoted by d(u;v). For all other notions and definitions, not given here, see, for example, (1), or (4) (for graph spectra). For the basic notions and terminology on the spectral graph theory the readers are referred to (4). Let A(G) be the adjacency matrix of G. Its eigenvalues are called the eigenvalues

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Bounds on normalized Laplacian eigenvalues of graphs

2014-08-21

Jianxi Li , Ji-Ming Guo , Wai Chee Shiu

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A Note on Laplacian Spectral Radius of Bipartite Graphs With Given Connectivity

2014-07-25

Miaomiao Han , Xiying Yuan , Jianxi Li

The number of spanning trees of composite graphs

2014-05-01

Jianxi Li , Wai Chee Shiu

Six classes of trees with largest normalized algebraic connectivity

2014-04-23

Jianxi Li , Ji-Ming Guo , Wai Chee Shiu +1 more

Coauthor	Papers Together
Wai Chee Shiu	38
Ji-Ming Guo	11
Yirong Zheng	9
Hongzhang Chen	8
Jian-Bo Lv	8
An Chang	8
Sarula Chang	7
Shou‐Jun Xu	6
Peng Huang	6
Wai Hong Ronald Chan	5
Ji-Ming Guo	4
Ji‐Ming Guo	4
Xiaoyun Lv	3
Dongxin Tu	2
Longzhe Jin	2
Yang Liu	2
Kaiyang Lan	1
Ş. Burcu Bozkurt Altındağ	1
Kexiang Xu	1
Lingping Zhong	1
Feng Liu	1
Xiying Yuan	1
An Chang	1
Shaoqiang Liu	1
L.V. Jian-Bo	1
Durmuş Bozkurt	1
Shiu Chee	1
Ho-Kuen Ng	1
Zhi-Wen Wang	1
Bit-Shun Tam	1
S. K. Ayyaswamy	1
Miaomiao Han	1
Nian Hong Zhou	1
Y. B. Venkatakrishnan	1
Sa Rula	1
Xiaoyun Lv	1
Ziwen Huang	1
Yuan Hou	1
Selvaraj Balachandran	1
Chee Shiu	1
E. K. S. LIU	1
Yi-Ping Liang	1
Pan-Pan Tong	1
Gee-Choon Lau	1

Algebraic connectivity of graphs

1973-01-01

Miroslav Fiedler

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On the second largest Laplacian eigenvalues of graphs

2012-12-11

Jianxi Li , Ji-Ming Guo , Wai Chee Shiu

Merging the A-and Q-spectral theories

2017-01-01

Vladimir Nikiforov

Let G be a graph with adjacency matrix A(G), and let D(G) be the diagonal matrix of the degrees of G: The signless Laplacian Q(G) of G is defined as … Let G be a graph with adjacency matrix A(G), and let D(G) be the diagonal matrix of the degrees of G: The signless Laplacian Q(G) of G is defined as Q(G):= A(G) +D(G). Cvetkovic called the study of the adjacency matrix the A-spectral theory, and the study of the signless Laplacian{the Q-spectral theory. To track the gradual change of A(G) into Q(G), in this paper it is suggested to study the convex linear combinations A_ (G) of A(G) and D(G) defined by A? (G) := ?D(G) + (1 - ?)A(G), 0 ? ? ? 1. This study sheds new light on A(G) and Q(G), and yields, in particular, a novel spectral Tur?n theorem. A number of open problems are discussed.

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Characterization of molecular branching

1975-11-01

Milan Randić

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTCharacterization of molecular branchingMilan RandicCite this: J. Am. Chem. Soc. 1975, 97, 23, 6609–6615Publication Date (Print):November 1, 1975Publication History Published online1 May 2002Published inissue 1 November 1975https://pubs.acs.org/doi/10.1021/ja00856a001https://doi.org/10.1021/ja00856a001research-articleACS … ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTCharacterization of molecular branchingMilan RandicCite this: J. Am. Chem. Soc. 1975, 97, 23, 6609–6615Publication Date (Print):November 1, 1975Publication History Published online1 May 2002Published inissue 1 November 1975https://pubs.acs.org/doi/10.1021/ja00856a001https://doi.org/10.1021/ja00856a001research-articleACS PublicationsRequest reuse permissionsArticle Views2766Altmetric-Citations2514LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InRedditEmail Other access optionsGet e-Alertsclose Get e-Alerts

The Laplacian Spectrum of a Graph

1990-04-01

Robert Grone , Russell Merris , V. S. Sunder

Let G be a graph. The Laplacian matrix $L(G) = D(G) - A(G)$ is the difference of the diagonal matrix of vertex degrees and the 0-1 adjacency matrix. Various aspects … Let G be a graph. The Laplacian matrix $L(G) = D(G) - A(G)$ is the difference of the diagonal matrix of vertex degrees and the 0-1 adjacency matrix. Various aspects of the spectrum of $L(G)$ are investigated. Particular attention is given to multiplicities of integer eigenvalues and to the effect on the spectrum of various modifications of G.

On the second largest Laplacian eigenvalue of trees

2005-04-21

Ji‐Ming Guo

The Laplacian Spectrum of a Graph II

1994-05-01

Robert Grone , Russell Merris

Let G be a graph. Denote by $D( G )$ the diagonal matrix of its vertex degrees and by $A( G )$ its adjacency matrix. Then $L( G ) = … Let G be a graph. Denote by $D( G )$ the diagonal matrix of its vertex degrees and by $A( G )$ its adjacency matrix. Then $L( G ) = D( G ) - A( G )$ is the Laplacian matrix of G. The first section of this paper is devoted to properties of Laplacian integral graphs, those for which the Laplacian spectrum consists entirely of integers. The second section relates the degree sequence and the Laplacian spectrum through majorization. The third section introduces the notion of a d-cluster, using it to bound the multiplicity of d in the spectrum of $L( G )$.

Laplacian graph eigenvectors

1998-07-01

Russell Merris

Spectra of Graphs

2011-12-15

Andries E. Brouwer , Willem H. Haemers

The harmonic index for graphs

2011-10-03

Lingping Zhong

The Harmonic Index on Unicyclic Graphs.

2012-01-01

Lingping Zhong

Some eigenvalue properties in graphs (conjectures of Graffiti — II)

1993-02-01

Odile Favaron , Maryvonne Mahéo , Jean-François Saclé

On the harmonic index and the chromatic number of a graph

2013-05-08

Hanyuan Deng , Selvaraj Balachandran , S. K. Ayyaswamy +1 more

Eigenvalues of the Laplacian of a graph∗

1985-10-01

William N. Anderson , T. D. Morley

Let G be a finite undirected graph with no loops or multiple edges. We define the Laplacian matrix of G,Δ(G)by Δij= degree of vertex i and Δij−1 if there is … Let G be a finite undirected graph with no loops or multiple edges. We define the Laplacian matrix of G,Δ(G)by Δij= degree of vertex i and Δij−1 if there is an edge between vertex i and vertex j. In this paper we relate the structure of the graph G to the eigenvalues of A(G): in particular we prove that all the eigenvalues of Δ(G) are non-negative, less than or equal to the number of vertices, and less than or equal to twice the maximum vertex degree. Precise conditions for equality are given.

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On the Laplacian spectral radius of a tree

2003-03-25

Ji‐Ming Guo

An Interlacing Result on Normalized Laplacians

2004-01-01

Guantao Chen , George J. Davis , Frank J. Hall +3 more

Given a graph G, the normalized Laplacian associated with the graph G, denoted ${\cal L}$(G), was introduced by F. R. K. Chung and has been intensively studied in the last … Given a graph G, the normalized Laplacian associated with the graph G, denoted ${\cal L}$(G), was introduced by F. R. K. Chung and has been intensively studied in the last 10 years. For a k-regular graph G, the normalized Laplacian ${\cal L}$ (G) and the standard Laplacian matrix L(G) satisfy L(G) = k {\cal L} (G), and hence they have the same eigenvectors and their eigenvalues are directly related. However, for an irregular graph G, ${\cal L} (G) and L(G) behave quite differently, and the normalized Laplacian seems to be more natural. In this paper, Cauchy interlacing-type properties of the normalized Laplacian are investigated, and the following result is established. Let G be a graph, and let H = G - e, where e is an edge of G. Let $\lambda_1\geq \lambda_2 \geq \cdots \geq \lambda_{n}=0$ be the eigenvalues of ${\cal L}(G)$, and let $\theta_1\geq \theta_2 \geq \cdots \geq \theta_{n}$ be the eigenvalues of ${\cal L}(H)$. Then, $\lambda_{k-1} \geq \theta_{k}\geq \lambda_{k+1}$ for each k = 1, 2, 3, \dots, n, where $\lambda_{0} =2$ and $\lambda_{n+1}=0$. Applications are given for eigenvalues of graphs obtained from special graphs by adding or deleting a few edges. A short proof is given of the result that G is a graph with each component a nontrivial bipartite graph if and only if $2-\lambda$ is an eigenvalue of ${\cal L}(G)$ for each eigenvalue $\lambda$ of ${\cal L }(G)$.

Interlacing eigenvalues and graphs

1995-09-01

Willem H. Haemers

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A conjecture on the algebraic connectivity of connected graphs with fixed girth

2007-12-06

Ji‐Ming Guo

Nullity of a graph in terms of the dimension of cycle space and the number of pendant vertices

2016-08-06

Xiaobin Ma , Dein Wong , Fenglei Tian

On the Laplacian spectral radius of trees with fixed diameter

2006-07-27

Ji‐Ming Guo

The effect on the Laplacian spectral radius of a graph by adding or grafting edges

2005-09-20

Ji‐Ming Guo

The Laplacian spectral radius of some bipartite graphs

2007-12-06

Xiao Zhang , Heping Zhang

Upper bounds on the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi></mml:math>-forcing number of a graph

2014-09-20

David Amos , Yair Caro , Randy Davila +1 more

On the general sum-connectivity index of trees

2010-10-31

Zhibin Du , Bo Zhou , Nenad Trinajstić

The harmonic index of a graph

2014-10-01

Jianxi Li , Wai Chee Shiu

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The spanning k-trees, perfect matchings and spectral radius of graphs

2021-10-04

Dandan Fan , Sergey Goryainov , Xueyi Huang +1 more

A $k$-tree is a spanning tree in which every vertex has degree at most $k$. In this paper, we provide a sufficient condition for the existence of a $k$-tree in … A $k$-tree is a spanning tree in which every vertex has degree at most $k$. In this paper, we provide a sufficient condition for the existence of a $k$-tree in a connected graph with fixed order in terms of the adjacency spectral radius and the signless Laplacian spectral radius, respectively. Also, we give a similar condition for the existence of a perfect matching in a balanced bipartite graph with fixed order and minimum degree.

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On the Laplacian, signless Laplacian and normalized Laplacian characteristic polynomials of a graph

2013-09-01

Ji-Ming Guo , Jianxi Li , Wai Chee Shiu

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On the A-spectral radius of a graph

2018-03-21

Jie Xue , Huiqiu Lin , Shuting Liu +1 more

A note on Laplacian graph eigenvalues

1998-12-01

Russell Merris

Sum-connectivity index of molecular trees

2010-05-10

Rundan Xing , Bo Zhou , Nenad Trinajstić

On a conjecture for the sum of Laplacian eigenvalues

2012-01-06

Wang Shouzhong , Yufei Huang , Bo Liu

On the harmonic index and the matching number of a tree.

2014-01-01

Jian-Bo Lv , Jianxi Li

A bound on the algebraic connectivity of a graph in terms of the number of cutpoints

2000-03-01

Steve Kirkland

Let G be a graph on n vertices which has k cutpoints. For the case that 2≤k≤n/2, we prove tha the algebraic connectivity of G is at most , and … Let G be a graph on n vertices which has k cutpoints. For the case that 2≤k≤n/2, we prove tha the algebraic connectivity of G is at most , and we explicitly characterize the graphs attaining equality in this bound.

On the sum of the two largest Laplacian eigenvalues of trees

2014-06-23

Mei Guan , Mingqing Zhai , Yongfeng Wu

Abstract For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> </mml:math> , the sum of the two largest Laplacian eigenvalues of a tree T , an upper bound is obtained. Moreover, among … Abstract For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> </mml:math> , the sum of the two largest Laplacian eigenvalues of a tree T , an upper bound is obtained. Moreover, among all trees with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:math> vertices, the unique tree which attains the maximal value of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> </mml:math> is determined. MSC: 05C50.

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A Note on the Normalized Laplacian Spectra

2011-02-01

Honghai Li , Jiong-Sheng Li

Let $G$ be a connected graph and $\mathcal{L}$ be its normalized Laplacian matrix. Let $\lambda_1$ be the second smallest eigenvalue of $\mathcal{L}$. In this paper we studied the effect on … Let $G$ be a connected graph and $\mathcal{L}$ be its normalized Laplacian matrix. Let $\lambda_1$ be the second smallest eigenvalue of $\mathcal{L}$. In this paper we studied the effect on the second smallest normalized Laplacian eigenvalue by grafting some pendant paths.

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Toughness and spectrum of a graph

1995-09-01

A.E. Brouwer

Upper bounds for the sum of Laplacian eigenvalues of graphs

2012-02-07

Zhibin Du , Bo Zhou

THE ALGEBRAIC MULTIPLICITY OF THE NUMBER ZERO IN THE SPECTRUM OF A BIPARTITE GRAPH

1972-01-01

Dragoš Cvetković , I. M. Gutman

The ordering of unicyclic graphs with the smallest algebraic connectivity

2009-02-17

Ying Liu , Yue Liu

The ordering of trees and connected graphs by algebraic connectivity

2008-01-09

Jia‐Yu Shao , Ji‐Ming Guo , Haiying Shan

Minimum general sum-connectivity index of unicyclic graphs

2010-06-08

Zhibin Du , Bo Zhou , Nenad Trinajstić

Ordering (signless) Laplacian spectral radii with maximum degrees of graphs

2014-10-21

Muhuo Liu , Bo Liu , Cheng Bo

On a novel connectivity index

2009-01-05

Bo Zhou , Nenad Trinajstić

On the sum of the Laplacian eigenvalues of a tree

2011-02-25

Eliseu Fritscher , Carlos Hoppen , Israel Rocha +1 more

Bounding the sum of the largest Laplacian eigenvalues of graphs

2014-02-19

Israel Rocha , Vilmar Trevisan

We prove that Brouwer's conjecture holds for certain classes of graphs. We also give upper bounds for the sum of the largest Laplacian eigenvalues for graphs satisfying certain properties: those … We prove that Brouwer's conjecture holds for certain classes of graphs. We also give upper bounds for the sum of the largest Laplacian eigenvalues for graphs satisfying certain properties: those that contain a path or a cycle of a given size, graphs with a given matching number and graphs with a given maximum degree. Then we provide conditions for which these upper bounds are better than the previous known results.

The effect on the second smallest eigenvalue of the normalized Laplacian of a graph by grafting edges

2008-05-12

Honghai Li , Jiong-Sheng Li , Yi-Zheng Fan

Let G be a connected graph and be its normalized Laplacian matrix. Let λ1 be the second smallest eigenvalue of and f be its corresponding harmonic eigenfunction. In this article … Let G be a connected graph and be its normalized Laplacian matrix. Let λ1 be the second smallest eigenvalue of and f be its corresponding harmonic eigenfunction. In this article we investigate the properties of the harmonic eigenfunction f on blocks of G. Next we use this result to characterize the effect on the second smallest normalized Laplacian eigenvalue by grafting edges.